Blow-up of nonnegative solutions of an abstract semilinear heat equation with convex source

• Published in: arXiv.org
• Main Authors: Lenz, Daniel, Schmidt, Marcel, Zimmermann, Ian
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.05.2022
• Subjects: Cauchy problems; Continuity (mathematics); Linear operators; Operators (mathematics); Thermodynamics
• ISSN: 2331-8422

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation \(u' = Lu + f(u)\) in \(L^p(X,m)\) for \(p \in [1,\infty)\), where \((X,m)\) is a \(\sigma\)-finite measure space, \(L\) is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in \(L^p(X,m)\), and \(f\) is a strictly increasing, convex, continuous function on \([0,\infty)\) with \(f(0) = 0\) and \(\int_1^\infty 1/f < \infty\). Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by \(L\) and the reaction represented by \(f\) in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. In the process, we recover and extend some older as well as recent results in a unified framework.